On the functional properties of the confluent hypergeometric zeta-function

Abstract

A zeta-function associated with Kummer's confluent hypergeometric function is introduced as a classical Dirichlet series. An integral representation, a transformation formula, and relation formulas between contiguous functions and one generalization of Ramanujan's formula are given. The inverse Laplace transform of confluent hypergeometric functions is essentially used to derive the integral representation.

The exponential-type generating function of the Riemann zeta-function revisited

“…In recent works [16][17][18], we derived some functional properties of Bessel zetafunctions and a confluent hypergeometric zeta-function. The J -Bessel zeta-function appears in the Fourier series expansion of the Poincaré series attached to SL(2, Z) by the inverse Mellin transform.…”

“…Following the techniques employed in [16,17], we drive an integral representation of (1.1) via the inverse Laplace transform of T α e −1/T . The integral expression of ζ exp (s; λ; z) gives a transformation formula when z ∈ R or functional relation when…”

Some generating functions of the Riemann zeta function

“…In recent works [No1,No2] we introduced some new zeta-functions, Bessel zetafunctions and a confluent hypergeometric zeta-function, where we derived integral representations, transformation formulas, power series expansions involving the Riemann zeta-function, and recurrence formulas. The J-Bessel zeta-function introduced in [No1] 108 T. NODA appears in the Fourier series expansion of the Poincaré series attached to SL(2, Z) by applying the inverse Mellin transform.…”